Measuring Assets’ Returns

• Let’s begin by focusing on two major asset classes - stocks and bonds
• How have these assets performed historically?
  • What’s the best way to summarize their risk + return?

Measuring Assets’ Returns

• Why does a Treasury bill’s return not fall below zero?
• Is monthly return the best measure? What else could we do?
  • Let’s quickly refresh ourselves on how to measure returns.

Return Definitions

• The rate of return is also known as the “net return”
  \[ r_{t} = \frac{P_{t} + D_{t}}{P_{t-1}} - 1 \]
• Gross returns (sometimes \(R_{t}\)) is referred to as \(1+r_{t}\)
• The risk-free rate will be called \(r_{f}\)
  • How can I get the risk-free rate?
• Excess returns above the risk free rate are \(r_{e} = r - r_{f}\)

Return Defintions

A holding-period return of \(T\) years is

\[ r_{0,T} = (1+r_{1})(1+r_{2})\ldots (1+r_{T}) - 1 \]

• How you measure the holding period matters a ton!
• Recall the mutual fund experiment

Return Definitions

• How do we compare returns across different horizons?
  • Say I want to compare a window of cumulative returns 5 v 10 years.

Return Definitions

• Arithmetic average returns \[ \overline{r} = \frac{1}{T}(r_{1} + r_{2} + \ldots + r_{T}) \]
• Geometric average returns \[ r_{G} = \left[(1+r_{1})(1+r_{2})\ldots (1+r_{T})\right]^{\frac{1}{T}} - 1 = \big[\frac{P_{T}}{P_{0}}\big]^{\frac{1}{T}} - 1 \]
  • Arithmetic average is unbiased estimate of 1-period future returns (expected returns)
  • Geometric average is a measure of cumulative past performance

Return Definitions

• We calculate estimates of variance using squared deviations from arithmetic average returns \[ \sigma^{2}(r) = VAR(r) = \frac{1}{T}\bigg((r_{1}-\overline{r})^{2} + (r_{2}-\overline{r})^{2} + \ldots + (r_{T}-\overline{r})^{2})\bigg) \]
• Standard deviation is the square-root of variance: \[ \sigma(r) = SD(r) = \sqrt{\frac{1}{T}\bigg((r_{1}-\overline{r})^{2} + (r_{2}-\overline{r})^{2} + \ldots + (r_{T}-\overline{r})^{2})\bigg)} \]

Return Defintions

• Finally, covariance measures how two returns move together \[ \begin{aligned} \sigma_{i,j} = COV(r_{i}, r_{j}) = \frac{1}{T}\bigg((r_{1,i}-\overline{r}_{i})(r_{1,j}-\overline{r}_{j}) &+ (r_{2,i}-\overline{r}_{i})(r_{2,j}-\overline{r}_{j})\\ &+ \ldots + (r_{T,i}-\overline{r}_{i})(r_{T,j}-\overline{r}_{j})\bigg) \end{aligned} \]
• Correlations scale the covariance by standard deviations \[ \rho_{i,j} = \frac{\sigma_{i,j}}{\sigma_{i}\sigma_{j}} \]
• Excel provides functions AVERAGE,GEOMEAN,VAR, STDEV, and COVAR

Wanted: Information about risk and return

• To make investment decisions, we need to know
  • … the expected future returns
  • … the riskiness of future returns
• We turn to historical return data for these
  • A caveat…
  • The data give a pretty good sense of the risk, but expected returns are hard to measure.
  • Why?

Mean 1 (purple): 4.03

Mean 2 (yellow): 4.69

True distributions:

Mean 1: 4

Mean 2: 5

Asset Returns: Historical Record

• US returns high, but not an outlier

Asset Returns: Historical Record

• Average s.d. of returns is 23% (US 20%)
• What’s up with Italy, Germany, Japan?

High returns or survivorship bias? (Jorion and Goetzmann (1999))

High returns or survivorship bias? (Anarkulova et al. 2020)

Still waiting for data?

How can we forecast returns going forward?

Recall that

\[ r_{1} = \underbrace{\frac{D_{1}}{P_{0}}}_{\text{dividend yield}} + \underbrace{\frac{P_{1}}{P_{0}}}_{\text{growth}} \]

Also,

\[ \underbrace{\frac{D_{1}}{P_{0}}}_{\text{dividend yield}} = \text{earnings payout share} \times \frac{E_{1}}{P_{0}} \] where, \(E_{1}\) is earnings.

This implies \[ r_{1} = \text{firm's payout share} \times \frac{E_{1}}{P_{0}} + \text{growth} \]

The Gordon Model

\[ r_{1} = \text{payout share} \times \frac{E_{1}}{P_{0}} + \text{growth} \]

• Historically, payout has been around 50%, P/E about 25 and growth rate of about 4-5% (nominal)
  • What does that imply about nominal expected returns?

Shiller Price-Earnings Ratios

Shiller Price-Earnings Ratios

Other risk measures

Value at Risk (VaR)

• A measure of loss most frequently associated with extreme negative returns
• VaR is the quantile of a distribution below which lies q % of the possible values of that distribution
• The 5% VaR, commonly estimated in practice, tells us how bad returns (or losses) will be in the worst 5% of times

VaR under normality

• 5% VaR return is equal to E(r) - 1.645 X s.d.
  • 5% Value at Risk (VaR) represents the lower bound on the return’s 10% confidence interval
• Under non-normality, look at historic returns for 5% cutoff

Expected Shortfall (ES)

• Also called conditional tail expectation (CTE)
• More conservative measure of downside risk than VaR
  • VaR takes the highest return from the worst cases
  • ES takes an average return of the worst cases

How would you want to model risk?

Consider the Swiss Franc case:

Takeaways

• We still have a limited amount of high quality data to make inferences from, but…
• Historical data suggests a premium for risk in asset returns, but the size of the premium is up for debate
• If expected returns for the aggregate stock market with 80+ years of data are so imprecise, what are we to do with individual stocks?
• Need more than data to understand expected returns